Graphs and C*-AlgebrasBy Steve Kaliszewski of Arizona State
University A directed graph is a set of
points, or vertices, together with a set of arrows, or edges, which
connect
some of the vertices. Graphs have had various and significant
applications
throughout mathematics, physical science, and computer science. For
instance, to solve the famous Konigsberg Bridge Problem in 1736, Leonhard
Euler made a graph in which the vertices represented pieces of land, and
the edges represented the bridges between them. Using simple
observations
about graphs, he was able to conclude that there was no round trip which
used each bridge exactly once. More abstractly, graphs have
been used to encode the algebraic structure of mathematical objects such
as groups (via their Coxeter graphs) and C*-algebras (via Bratteli
diagrams).
Bratteli diagrams, for instance, provided a complete classification of
the C*-algebras known as AF-algebras. Reversing the trend of the
previous
examples (in which a graph was associated to a problem or object from
another
subject), it has been found that a C*-algebra can be associated to a
directed
graph in such a way that the structure of the graph is reflected in the
structure of the algebra. This development has injected
C*-theory with a vast array of interesting yet manageable and concrete
examples: a large class of C*-algebras, called the Cuntz-Krieger algebras,
for instance, can be realized as graph C*-algebras. A flourishing
industry --- much of it originating from ASU --- is now centered around
the study of C*-algebras arising from directed graphs, their
generalizations,
and their applications.The extent to which a C*-algebra can capture the
structure
of a graph is sometimes remarkable: for example, the C*algebra of any
finite
graph with no directed loops is built out of basic (matrix) algebras in
a very natural and predictable way. Other graph constructions are
also reflected in analogous C*-algebraic constructions: for instance, the
C*-algebra of a Cartesian product of graphs is a tensor product of the
individual graph C*-algebras.ASU Professors Steve Kaliszewski and John Quigg,
together with Professor Iain Raeburn of the University of Newcastle,
Australia,
have recently extended the "dictionary" which translates between graph
concepts and C*-algebraic concepts to include several more fundamental
constructions. Strikingly, these constructions had arisen
independently
of one another in graph theory and in C*-theory, and therefore appeared
superficially to be so different that the analogies between them had not
yet been noticed. Besides having significant applications in
C*-theory,
results like these are expected to provide new and useful tools for graph
theorists. Because of the widespread applications of graphs, this
in turn could have repercussions throughout much of mathematics, physical
science, and computer science.